MCS 477 AAMU Numerical Analysis Lipschitz Constant and Initial Value Problems

Problem 1: Verify that y(t) = Ce?t + t2 ? 2t + 2 solves y0 = t2 ? y and find aLipschitz constant L for the rectangle R = { (t, y) : 0 ? t ? 3, 0 ? y ? 5 }.

Problem 2: Show that by solving the initial value problem y0 = f(t) for a ? t ? bwith y(a) = 0, it is possible to compute R baf(t)dt.

Problem 3: (Computer problem. You may use code from the textbook.) Considerthe initial value problemy0 =t ? y2on [0, 3] with y(0) = 1.The exact solution is y(t) = 3e?t/2 ? 2 + t. Use Euler’s method with h = 1, 1/2, 1/4and 1/8 to approximate y(t). Graph and label the approximations and the exactsolution on the same graph.

Problem 4: Consider the initial value problemy0 = t2 ? y with y(0) = 1.The exact solution is y(t) = ?e?t +t2 ?2t+ 2. Use the Runge-Kutta method of order4 (formula (6.50) of the textbook) for hand calculations with h = 0.2 and h = 0.1 toapproximate y(0.4). Does the error behave as expected?

Problem 5: For the following Linear Multistep Methods, determine whether theyare convergent, consistent, stable. (Three answers in each case are expected)(1) wi+1 =12wi +12wi?1 + 2hf(wi)(2) wi+1 = wi.

Problem 6 : Consider the boundary value problem ?u”(x) + u(x) = ?2 on(0, 1) with u(0) = 1 and u(1) = 0. Divide [0, 1] into four subintervals of equal sizeand apply the method of finite differences to set up the system of equations for theunknown values